exponential dispersion model

There are two versions to formulate an exponential dispersion model.

In the univariate case, a real-valued random variable $X$ belongs to the additive exponential dispersion model with canonical parameter $\backslash theta$ and index parameter $\backslash lambda$, $X\; \backslash sim\; \backslash mathrm\{ED\}^*(\backslash theta,\; \backslash lambda)$, if its probability density function can be written as

:$f\_X(x\backslash mid\backslash theta,\; \backslash lambda)\; =\; h^*(\backslash lambda,x)\; \backslash exp\backslash left(\backslash theta\; x\; -\; \backslash lambda\; A(\backslash theta)\backslash right)\; \backslash ,\backslash !\; .$

The distribution of the transformed random variable $Y=\backslash frac\{X\}\{\backslash lambda\}$ is called reproductive exponential dispersion model, $Y\; \backslash sim\; \backslash mathrm\{ED\}(\backslash mu,\; \backslash sigma^2)$, and is given by

:$f\_Y(y\backslash mid\backslash mu,\; \backslash sigma^2)\; =\; h(\backslash sigma^2,y)\; \backslash exp\backslash left(\backslash frac\{\backslash theta\; y\; -\; A(\backslash theta)\}\{\backslash sigma^2\}\backslash right)\; \backslash ,\backslash !\; ,$

with $\backslash sigma^2\; =\; \backslash frac\{1\}\{\backslash lambda\}$ and $\backslash mu\; =\; A\text{'}(\backslash theta)$, implying $\backslash theta\; =\; (A\text{'})^\{-1\}(\backslash mu)$.

The terminology dispersion model stems from interpreting $\backslash sigma^2$ as dispersion parameter. For fixed parameter $\backslash sigma^2$, the $\backslash mathrm\{ED\}(\backslash mu,\; \backslash sigma^2)$ is a natural exponential family.

In the multivariate case, the n-dimensional random variable $\backslash mathbf\{X\}$ has a probability density function of the following form

:$f\_\{\backslash mathbf\{X\}\}(\backslash mathbf\{x\}|\backslash boldsymbol\{\backslash theta\},\; \backslash lambda)\; =\; h(\backslash lambda,\backslash mathbf\{x\})\; \backslash exp\backslash left(\backslash lambda(\backslash boldsymbol\backslash theta^\backslash top\; \backslash mathbf\{x\}\; -\; A(\backslash boldsymbol\backslash theta))\backslash right)\; \backslash ,\backslash !,$

where the parameter $\backslash boldsymbol\backslash theta$ has the same dimension as $\backslash mathbf\{X\}$.

The cumulant-generating function of $Y\backslash sim\backslash mathrm\{ED\}(\backslash mu,\backslash sigma^2)$ is given by

:$K(t;\backslash mu,\backslash sigma^2)\; =\; \backslash log\backslash operatorname\{E\}[e^\{tY\}]\; =\; \backslash frac\{A(\backslash theta+\backslash sigma^2\; t)-A(\backslash theta)\}\{\backslash sigma^2\}\backslash ,\backslash !\; ,$

with $\backslash theta\; =\; (A\text{'})^\{-1\}(\backslash mu)$

Mean and variance of $Y\backslash sim\backslash mathrm\{ED\}(\backslash mu,\backslash sigma^2)$ are given by

:$\backslash operatorname\{E\}[Y]=\; \backslash mu\; =\; A\text{'}(\backslash theta)\; \backslash ,,\; \backslash quad\; \backslash operatorname\{Var\}[Y]\; =\; \backslash sigma^2\; A\text{'}\text{'}(\backslash theta)\; =\; \backslash sigma^2\; V(\backslash mu)\backslash ,\backslash !\; ,$

with unit variance function $V(\backslash mu)\; =\; A\text{'}\text{'}((A\text{'})^\{-1\}(\backslash mu))$.

If $Y\_1,\backslash ldots,\; Y\_n$ are i.i.d. with $Y\_i\backslash sim\backslash mathrm\{ED\}\backslash left(\backslash mu,\backslash frac\{\backslash sigma^2\}\{w\_i\}\backslash right)$, i.e. same mean $\backslash mu$ and different weights $w\_i$, the weighted mean is again an $\backslash mathrm\{ED\}$ with

:$\backslash sum\_\{i=1\}^n\; \backslash frac\{w\_i\; Y\_i\}\{w\_\{\backslash bullet\}\}\; \backslash sim\; \backslash mathrm\{ED\}\backslash left(\backslash mu,\; \backslash frac\{\backslash sigma^2\}\{w\_\backslash bullet\}\backslash right)\; \backslash ,\backslash !\; ,$

with $w\_\backslash bullet\; =\; \backslash sum\_\{i=1\}^n\; w\_i$. Therefore $Y\_i$ are called reproductive.

The probability density function of an $\backslash mathrm\{ED\}(\backslash mu,\; \backslash sigma^2)$ can also be expressed in terms of the unit deviance $d(y,\backslash mu)$ as

:$f\_Y(y\backslash mid\backslash mu,\; \backslash sigma^2)\; =\; \backslash tilde\{h\}(\backslash sigma^2,y)\; \backslash exp\backslash left(-\backslash frac\{d(y,\backslash mu)\}\{2\backslash sigma^2\}\backslash right)\; \backslash ,\backslash !\; ,$

where the unit deviance takes the special form $d(y,\backslash mu)\; =\; y\; f(\backslash mu)\; +\; g(\backslash mu)\; +\; h(y)$ or in terms of the unit variance function as $d(y,\backslash mu)\; =\; 2\; \backslash int\_\backslash mu^y\backslash !\; \backslash frac\{y-t\}\{V(t)\}\; \backslash ,dt$.

Many very common probability distributions belong to the class of EDMs, among them are: normal distribution, binomial distribution, Poisson distribution, negative binomial distribution, gamma distribution, inverse Gaussian distribution, and Tweedie distribution.

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Category:Statistical models